According to the TARDIS Data Core there was about two or three years between the appearance of the 9th doctor and his regeneration. From the same source we can estimate that the 10th doctor lived around seven or eight years.

However the doctor claimed to have lived in his 11th incarnation for around three hundred years!

Why would Matt Smith's incarnation of the doctor live so much longer than his 9th or 10th, given Matt Smith and David Tennant portrayed the character for roughly the same number of episodes?

2 Answers
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Different question: Why shouldn't it be the case, i.e. have the same timespan per incarnation?

Don't forget that all these three regenerations weren't triggered due to any natural/inherent reason (like dieing of old age). The ninth Doctor absorbed the vortex energy from Rose and the tenth Doctor absorbed the radiation from that completely weird and odd failsafe containment thingy. Neither of them regenerated by choice.

Also keep in mind that timespans are hardly constant when comparing single episodes. While this is especially true for time travelling stuff, it's also true for pretty much any other TV series or movie (maybe except 24).

For example, take episodes such as Rose or Aliens of London. The story happening in those essentially covers a day or two each. Compare this to episodes with far longer timespans, e.g. The Impossible Astronaut and the sequel: Those cover several months. Or, if you want it even more time bending: Blink happens on a single day for the audience as well as for the characters depicted in the present. However, for the Doctor (and Martha) obviously months pass.

Firstly, most of 11's life was spent at the battle of Trenzalore, he spent at least a couple hundred years there.
Secondly, there was his moody period where he just shut himself away for a very long time (on top of a fake cloud with a very long ladder, as you do).

Then there's the fact that he spent a particularly long time searching for Clara before settling down as a monk.
Then of course there's all those adventures he had with River.